scholarly journals Oscillation of a Class of Fractional Differential Equations with Damping Term

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Huizeng Qin ◽  
Bin Zheng
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Riccati Transformation ◽  
Damping Term ◽  
Variable Transformation ◽  
Average Technique ◽  
Integer Order ◽  
Fractional Differential

We investigate the oscillation of a class of fractional differential equations with damping term. Based on a certain variable transformation, the fractional differential equations are converted into another differential equations of integer order with respect to the new variable. Then, using Riccati transformation, inequality, and integration average technique, some new oscillatory criteria for the equations are established. As for applications, oscillation for two certain fractional differential equations with damping term is investigated by the use of the presented results.

scholarly journals Interval Oscillation Criteria for a Class of Fractional Differential Equations with Damping Term

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Chunxia Qi ◽  
Junmo Cheng
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Riccati Transformation ◽  
Damping Term ◽  
Oscillation Criteria ◽  
Interval Oscillation ◽  
Inequality Technique ◽  
Fractional Differential

Some new interval oscillation criteria are established based onthe certain Riccati transformation and inequality techniquefor a class of fractional differential equations with damping term. For illustrating the validity of the established results, we also present some applications for them.

Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

2020 ◽  
Vol 21 (6) ◽  
pp. 571-587 ◽  
Author(s):  
Akbar Zada ◽  
Sartaj Ali ◽  
Tongxing Li
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Integer Order ◽  
New Class ◽  
Proposed Model ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

scholarly journals Numerical Method for Fractional-order Problems Using MATLAB software

2020 ◽  
pp. 437-441
Author(s):  
Aye Mya Mya Moe ◽  
Aye Thida Myint ◽  
Zin Nwe Khaing
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Memory Term ◽  
Scalar Case ◽  
Implicit Methods ◽  
Efficient Treatment ◽  
Integer Order ◽  
Persistent Memory ◽  
Fractional Differential

Solving of Fractional differential equations of fractional (i.e., non-integer) order in an accurate, reliable and efficient way is much more difficult than in the standard integer-order case; moreover, the majority of the computational tools do not provide built-in functions for this kind of problem. In this paper was included the effective families of numerical methods for fractional-order problems, and the major computational issues such as the efficient treatment of the persistent memory term and the solution of the nonlinear systems involved in implicit methods using MATLAB routines specifically devised for solving three families of fractional-order problems: fractional differential equations (FDEs) (also for the non-scalar case), multi-order systems (MOSs) of FDEs and multi-term FDEs (also for the non-scalar case); some examples are provided to illustrate the use of the routines.

Linear stationary fractional differential equations

2019 ◽  
Vol 22 (2) ◽  
pp. 412-423
Author(s):  
Valeriy Nosov ◽  
Jesús Alberto Meda-Campaña
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Integer Order ◽  
Fractional Differential ◽  
Stability Results

Abstract In this paper, fractional-order derivatives satisfying conventional concepts, are considered in order to present some stability results on linear stationary differential equations of fractional-order. As expected, the obtained results are very close to the ones widely accepted in differential equations of integer order. Some examples are included in order to show how some restrictions of more sophisticated fractional-order derivatives are overcome.

scholarly journals Some New Oscillation Criteria for a Class of Nonlinear Fractional Differential Equations with Damping Term

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Damping Term ◽  
Oscillation Criteria ◽  
Nonlinear Fractional Differential Equations ◽  
Inequality Technique ◽  
Fractional Differential ◽  
Oscillation Of Solutions

We are concerned with oscillation of solutions of a class of nonlinear fractional differential equations with damping term. Based on a generalized Riccati function and inequality technique, we establish some new oscillation criteria for it. Some applications are also presented for the established results.

On the Cole–Hopf transformation and integration by parts formulae in computational methods within fractional differential equations and fractional optimal control theory

2021 ◽  
pp. 107754632110310
Author(s):  
Ahmed S Hendy ◽  
Mahmoud A Zaky ◽  
José A Tenreiro Machado
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Heat Equation ◽  
Fractional Differential Equations ◽  
Burgers Equation ◽  
Variable Order ◽  
Integer Order ◽  
Order Variable ◽  
Fractional Differential ◽  
Distributed Order

The treatment of fractional differential equations and fractional optimal control problems is more difficult to tackle than the standard integer-order counterpart and may pose problems to non-specialists. Due to this reason, the analytical and numerical methods proposed in the literature may be applied incorrectly. Often, such methods were established for the classical integer-order operators and are then applied directly without having in mind the restrictions posed by their fractional-order versions. It was recently reported that the Cole–Hopf transformation can be used to convert the time-fractional nonlinear Burgers’ equation into the time-fractional linear heat equation. In this article, we show that, unlike integer-order differential equations, employing the Cole–Hopf transformation for reducing the nonlinear time-fractional Burgers’ equation into the time-fractional heat equation is wrong from two different perspectives. Indeed, such a reduction is accomplished using incorrect transcripts of the Leibniz or chain rules. Hence, providing numerical or analytical schemes based on the Cole–Hopf transformation leads to erroneous results for the nonlinear time-fractional Burgers’ equation. Regarding constant-order, variable-order, and distributed-order Caputo fractional optimal problems, we note an inconsistency in the necessary optimality conditions derived in the literature. The transversality conditions were introduced as identical to those for the integer-order case, with a vanishing multiplier at the terminal of the interval. The correct condition should involve a constant-order, variable-order, or distributed-order fractional integral operator. We also deduce that if the control system is defined with a Caputo derivative, then the adjoint equations should be expressed in the Riemann–Liouville sense and vice versa. In fact, neglecting some terms in the integration by parts formulae, during the derivation of the optimality conditions, causes some confusion in the literature.

A Numerical Method for Caputo Differential Equations and Application of High-Speed Algorithm

2019 ◽  
Vol 14 (9) ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
High Speed ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Computing Time ◽  
Integer Order ◽  
Truncation Errors ◽  
Fractional Differential

In this paper, a numerical algorithm to solve Caputo differential equations is proposed. The proposed algorithm utilizes the R2 algorithm for fractional integration based on the fact that the Caputo derivative of a function f(t) is defined as the Riemann–Liouville integral of the derivative f(ν)(t). The discretized equations are integer order differential equations, in which the contribution of f(ν)(t) from the past behaves as a time-dependent inhomogeneous term. Therefore, numerical techniques for integer order differential equations can be used to solve these equations. The accuracy of this algorithm is examined by solving linear and nonlinear Caputo differential equations. When large time-steps are necessary to solve fractional differential equations, the high-speed algorithm (HSA) proposed by the present authors (Fukunaga, M., and Shimizu, N., 2013, “A High Speed Algorithm for Computation of Fractional Differentiation and Integration,” Philos. Trans. R. Soc., A, 371(1990), p. 20120152) is employed to reduce the computing time. The introduction of this algorithm does not degrade the accuracy of numerical solutions, if the parameters of HSA are appropriately chosen. Furthermore, it reduces the truncation errors in calculating fractional derivatives by the conventional trapezoidal rule. Thus, the proposed algorithm for Caputo differential equations together with the HSA enables fractional differential equations to be solved with high accuracy and high speed.

scholarly journals Oscillation for a Class of Right Fractional Differential Equations on the Right Half Line with Damping

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Hui Liu ◽  
Run Xu
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Half Line ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation ◽  
Fractional Differential ◽  
The Right

In this paper, we discuss a class of fractional differential equations of the form D-α+1y(t)·D-αy(t)-p(t)f(D-αy(t))+q(t)h∫t∞(s-t)-αy(s)ds=0.D-αy(t) is the Liouville right-sided fractional derivative of order α∈(0,1). We obtain some oscillation criteria for the equation by employing a generalized Riccati transformation technique. Some examples are given to illustrate the significance of our results.

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